The directed power graph $\vec{\mathcal P}(\mathbf G)$ of a group $\mathbf G$ is the simple digraph with vertex set $G$ such that $x\rightarrow y$ if $y$ is a power of $x$. The power graph of $\mathbf G$, denoted by $\mathcal P(\mathbf G)$, is the underlying simple graph. The enhanced power graph $\mathcal P_e(\mathbf G)$ of $\mathbf G$ is the simple graph with vertex set $G$ in which two elements are adjacent if they generate a cyclic subgroup. In this paper, it is proven that, if two groups have isomorphic power graphs, then they have isomorphic enhanced power graphs, too. It is known that any finite nilpotent group of order divisible by at most two primes has perfect enhanced power graph. We investigated whether the same holds for all finite groups, and we have obtained a negative answer to that question. Further, we proved that, for any $n\geq 0$ and prime numbers $p$ and $q$, every group of order $p^nq$ and $p^2q^2$ has perfect enhanced power graph. We also give a complete characterization of symmetric and alternative groups with perfect enhanced graphs.

Theorem 4.4 of the first version of the manuscript turned out to be incorrect, which we show in the present version by providing a counterexample. Therefore, Theorem 4.4 and Corollary 4.5 of the first version have been removed. Further, Theorems 4.8 and 4.9 have been added. Also, the proof of Proposition 4.11 has been changed as it relied on Theorem 4.4 of the first version